Coupling in Magnetic Multilayers M.A

Anisotropy of Neel “orange-peel” coupling in magnetic multilayers 1,3 1,2,* 1,3 M.A. Kuznetsov , O.G.Udalov , A.A.Fraerman 1Institute for physics of microstructures RAS, 7 Academicheskaya Str., Afonino, Nizhny Novgorod region, Kstovsky district, Kstovo region, 603087, Russia 2California State University Northridge, 18111 Nordhoff Street, Northridge, CA 91330, USA 3Lobachevsky State University of Nizhni Novgorod, 23 Prospekt Gagarina (Gagarin Avenue), Nizhny Novgorod, 603950, Russia * Corresponding author e-mail: [email protected] Highlights: - “Orange-peel” coupling between two ferromagnetic layers is calculated for an arbitrary orientation of the layers magnetizations - “Orange-peel” coupling changes sign when the magnetizations rotate from in-plane orientation to out-of-plane orientation Abstract: We calculate the energy of the magnetostatic interaction between two ferromagnetic films with uniform magnetization and correlated interfaces (the “orange-peel” effect). The “orange-peel” coupling is anisotropic: the interaction is ferromagnetic when the films are magnetized in-plane; and it is antiferromagnetic when magnetization is out-of-plane. The interaction anisotropy can be used to distinguish the “orange-peel” effect from the interlayer exchange coupling. Key words: orange peel, ferromagnets, interlayer coupling The magnetostatic interaction of two magnetic layers appears due to the roughness of their interfaces. The roughness produces a stray field. This field acts on the neighboring film leading to the so-called “orange-peel” interlayer interaction [1-8]. Often the “orange-peel” (OP) coupling is comparable to the interlayer exchange interaction [5, 9-14], which appears due to spin currents of conduction electrons [15]. These two interaction mechanisms should be distinguished in certain cases. This can be done by studying the angular dependencies of the interlayer coupling energy. The exchange interaction is isotropic and depends on the mutual orientation of magnetizations 푊ex~(퐌1퐌2), where 퐌1,2 are layers magnetic moments. In the present manuscript we will show that “orange-peel” interaction is anisotropic and obeys the relation 푊op~퐽⊥푀1⊥푀2⊥ − 퐽||푀1||푀2||, where 푀1,2⊥ are the out-of-plane magnetizations components, 푀1,2|| are the in-plane components, 퐽⊥ and 퐽|| are positive and of the same order. Consider a system of two ferromagnetic films with rough surfaces (see Fig. 1). We assume that these surfaces are correlated and can be described by the same function 휉(훒), where 훒 = (푥, 푦) is the in-plane coordinate vector. Often the correlation between the film interfaces naturally appears during the films growth [16]. Therefore, this assumption is quite relevant. The ferromagnetic films are separated by a non-magnetic spacer of thickness 푎. We also assume that both layers are uniformly 1 magnetized. Their magnetizations are defined as follows 퐌1 = 푀1(cos(훼),0, sin(훼)), 퐌2 = 푀2(cos(훽) , 0, sin(훽)). The thicknesses of the layers are ℎ1and ℎ2. Fig. 1. Two rough ferromagnetic layers separated by a non-magnetic spacer. ℎ1 and ℎ2 denote the thicknesses of the magnetic layers. Spacer thickness is 푎. Horizontal lines show average interfaces. 퐌1,2 are the magnetizations of the ferromagnetic layers. 훼 and 훽 are the angles between 퐌1,2 and the average interfaces. 휉(푥, 푦) is the function describing the system roughness. The magnetostatic energy is given by 푧=ℎ1+푎+ℎ2+휉(훒) 푊op = − 푀2 ∫ 푑푥푑푦 [∫ (cos (훽) ∙ 퐻푥 + sin (훽) ∙ 퐻푧)푑푧], (1) 푧=ℎ1+푎+휉(훒) where 퐻푥,푧 are the x- and z-components of the magnetic field produced by the first layer in the region of the second layer. The field can be found using Maxwell equations div퐁 = 0, rot퐇 = 0. (2) Introducing the magnetic scalar potential 휑 (퐇 = −∇휑) one can transform Eqs. (2) to Δ휑 = 0. (3) At the interfaces 푧 = ℎ1 + 휉 and 푧 = 휉 we require that the normal component of the magnetic induction 퐁 and the scalar potential 휑 are continuous (퐼) (퐼퐼) −(퐧∇휑 ) = −(퐧∇휑 ) + 4휋(퐌1퐧)|푧=ℎ1+휉 (4) (퐼퐼퐼) (퐼퐼) −(퐧∇휑 ) = −(퐧∇휑 ) + 4휋(퐌1퐧)|푧=휉 (5) (퐼) (퐼퐼) (퐼퐼) (퐼퐼퐼) 휑 = 휑 |푧=ℎ1+휉 , 휑 = 휑 |푧=휉. (6) Here 휑(퐼퐼) is the potential inside the first ferromagnetic film, notations 휑(퐼) and 휑(퐼퐼퐼) are used for the potential above (푧 > ℎ1 + 휉) and below (푧 < 휉) the film, 퐧 is the local normal given by 2 휉푥 휉푦 1 휕휉 휕휉 퐧 = ( , , − ) , 휉푥 = , 휉푦 = . (7) 2 2 2 2 2 2 휕푥 휕푦 √1+휉푥+휉푦 √1+휉푥+휉푦 √1+휉푥+휉푦 We assume that the characteristic roughness height 휉0 is small comparing to the characteristic lengthscale 퐿 of the roughness in the (x,y)-plane, 휉0 ≪ 퐿. Then we can express the scalar potential as a series 휑 = 휑0 + 휑1 + …, (8) 푛 in which 휑푛~휉0 (푛 > 1). The potential 휑0 is given by 2휋푀1ℎ1 sin(훼) , 푧 > ℎ1 + 휉, region (퐼), ℎ 휑 = {4휋푀 (푧 − 1)sin(훼), 휉 < 푧 < ℎ + 휉, region (퐼퐼), (9) 0 1 2 1 −2휋푀1ℎ1 sin( 훼) , 푧 < 휉, region (퐼퐼퐼). Note that 휑0 is not the solution for the case of a flat surface (휉 = 0). We choose the potential 휑0 pursuing two goals. At first, far from the interface (푧 > 휉) the potential should turn into the solution for the flat interface. The second one is that the 휑0 should be split into two regions along the true interface. This makes solution of the boundary condition equations much easier. Choosing the zero- order potential as the solution for the perfectly flat surface requires reducing of the boundary conditions to the flat surface as well. While this is also possible, the procedure considered here looks more relevant. 2 The non-zero magnetostatic interaction energy appears in the second order in 휉0 (푊op~휉0 ), therefore one can calculate the scalar potential up to the first order in 휉0 . In the zero order one has Introducing Eqs. (8) and (9) into Eqs. (3-6) one gets the following equations for 휑1 (퐼) (퐼퐼) (퐼퐼퐼) (퐼퐼) 휑1 = 휑1 + 4 휋푀1휉 sin( 훼) |푧=ℎ1 , 휑1 = 휑1 + 4 휋푀1 휉sin( 훼) |푧=0, (10) (퐼) (퐼퐼) 휕휑 휕휑 1 = 1 + 4휋푀 휉 cos(훼) | , (11) 휕푧 휕푧 1 푥 푧=ℎ1 (퐼퐼퐼) (퐼퐼) 휕휑 휕휑 1 = 1 + 4휋푀 휉 cos( 훼) | . (12) 휕푧 휕푧 1 푥 푧=0 We introduce the in-plane Fourier components of the potential as follows 휑1(퐤, z) = −퐤훠 ∫ 휑1(훒, z)푒 푑푥푑푦. The z-dependence of the Fourier components can be expressed in the form −푘푧 퐴(퐤)푒 , 푧 > ℎ1 + 휉, region (퐼), −푘푧 푘푧 휑1(퐤) = {퐵(퐤)푒 + 퐶(퐤)푒 , 휉 < 푧 < ℎ1 + 휉, region (퐼퐼), (13) 퐷(퐤)푒푘푧, 휉 < 푧, region (퐼퐼퐼), where 푘 = |퐤| is the absolute value of the wavevector. The coefficients 퐴, 퐵, 퐶 and 퐷 in Eq. (13) are defined by the boundary conditions Eqs. (10) – (12). Note that the potential 휑1(퐤) should be expanded into a series with respect to 푘휉0 ∼ 휉0/퐿 around the lines 푧 = ℎ1 and 푧 = 0 when introducing into Eqs. (10-12). Only the lowest term should be conserved. The terms of higher order in 푘휉0 will contribute to 3 the higher order corrections 휑2, 휑3 etc. Since we are interested in the potential in the region of the second magnetic film (region (I)) we provide here only the expression for the coefficient 퐴 푘ℎ1 푘ℎ1 푘푥 퐴(퐤) = 4휋푀 푒 2 푆ℎ ( ) 휉(퐤) (sin( 훼) − cos(훼)), (14) 1 2 푘 ( ) 1 휑 퐼 (훒) = ∫ 퐴(퐤)푒−푘푧푒풌흆푑푘 푑푘 (15) 1 (2휋)2 푥 푦 For the interaction energy one has 푊op = −퐽푥 cos 훼 cos 훽 + 퐽푧 sin 훼 sin 훽, (16) 1 푘2 퐽 = 푀 푀 ∫ 푒−푘푎(1 − 푒−푘ℎ1)(1 − 푒−푘ℎ2) 푥 휉(퐤) 휉(−퐤)푑푘 푑푘 (17) 푥 2휋 1 2 푘 푥 푦 1 퐽 = 푀 푀 ∫ 푒−푘푎(1 − 푒−푘ℎ1)(1 − 푒−푘ℎ2)푘휉(풌) 휉(−풌)푑푘 푑푘 (18) 푧 2휋 1 2 푥 푦 2 Since 휉(퐤)휉(−퐤) = |휉(퐤)| , both coefficients in Eq. (16) are positive 퐽푥, 퐽푧 > 0. Note that the correlation of magnetic films interfaces plays a crucial role when performing integration in Eq. (1). The result Eq. (16) appears due to the correlation between the magnetization distribution in the second film and the magnetic field produced by the first layer. If the interfaces were not correlated, the OP effect would be zero. The interaction strength depends on the thickness of the magnetic layers and on the thickness of the insulating spacer. If the thickness of the magnetic layers exceeds the characteristic in-plane length scale of roughness 푑1,2 ≫ 퐿 the interaction does not depend on 푑1,2. This is quite clear since the stray field created by the roughness decays within the distance 퐿 from the interface. When the magnetic layers are thin 푑1,2 ≪ 퐿 the interaction strength linearly grows with increasing of the layer thicknesses 퐽 ∼ 푑1푑2. This is because the interaction volume grows with 푑1,2 in this limit. Note that in most experimental magnetic multilayer systems of interest the film thickness can be rather small. Therefore, the limit of thin films is more relevant. The exponential decay of the “orange-peel” interaction on the insulator thickness is the same as was obtained in a seminal paper of Neel [1] and Ref. [5]. Equations (17) and (18) show that the magnetostatic interaction energy is anisotropic. The anisotropy means that the coupling strength depends on the angle between the layers magnetic moments and the average surface normal (z-axis in our case). Moreover, when the magnetizations are in the (x,y)-plane the interaction is ferromagnetic, while when 퐌1,2 are along z-axis the interaction is antiferromagnetic.

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